Minkowski's Inequality for Integrals

The following inequality is a generalization of Minkowski's inequality C12.4 to double integrals. In some sense it is also a theorem on the change of the order of iterated integrals, but equality is only obtained if p = 1. 13.14 Theorem (Minkowski's inequality for integrals) Let XX and YY be-finite measure spaces and u u X × Y → ¯ be ⊗-measurable. Then X Y uuxx yy dyy p dxx 1/p Y X uuxx yy p dxx 1/p dyy holds for all p ∈ 1 , with equality for p = 1. Proof If p = 1, the assertion follows directly from Tonelli's theorem 13.8. If p > 1 we set U k xx = Y uuxx yy dyy ∧ k 1 A k xx where A k ∈ is a sequence with A k ↑ X and A k <. Without loss of generality we may assume that U k xx > 0 on a set of positive-measure, otherwise the left-hand side of the above inequality would be 0 (using Beppo Levi's theorem 9.6) and there would be nothing to prove. By Tonelli's theorem and Hölder's inequality T12.2 with 1 p +